Molecular Kinetic Energy and Temperature

In the previous lesson, we derived the key result linking kinetic theory and the ideal gas equation. Now we explore its consequences in depth: the relationship between molecular kinetic energy and temperature, the concept of root mean square speed, and the distribution of molecular speeds described by the Maxwell-Boltzmann distribution.

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The Mean Kinetic Energy of a Molecule

From kinetic theory and the ideal gas equation, we found:

⅓m⟨c²⟩ = kT

Multiplying both sides by 3/2:

½m⟨c²⟩ = 3/2 kT

This is one of the most important equations in thermal physics. It states that:

The mean translational kinetic energy of a single molecule is equal to 3/2 kT.

Key points:

• ½m⟨c²⟩ is the average kinetic energy per molecule (in joules).
• k = 1.38 × 10⁻²³ J K⁻¹ is the Boltzmann constant.
• T is the absolute temperature (in kelvin).
• This result is independent of the mass of the molecule and independent of the type of gas. At the same temperature, all ideal gas molecules have the same average kinetic energy — lighter molecules simply move faster.

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Root Mean Square Speed

The root mean square (rms) speed is defined as:

c_rms = √⟨c²⟩